Introduction
Cancer treatment increasingly relies on combination therapies to improve patient outcomes while reducing toxicity. Chemotherapy directly kills tumor cells, whereas immunotherapy enhances the body’s immune response. Determining optimal dosing schedules for both therapies remains a major challenge. This study develops an optimal control framework to investigate combined chemotherapy and immunotherapy strategies that minimize tumor burden while controlling treatment cost and side effects.

Methods
We consider a nonlinear system of ordinary differential equations describing interactions among tumor cells, immune cells, and chemotherapeutic drug concentration. Two time-dependent control variables are introduced: chemotherapy dosage and immunotherapy input. Pontryagin’s Maximum Principle is applied to derive necessary optimality conditions for minimizing an objective functional that balances tumor reduction, immune preservation, and treatment cost. The optimality system is solved numerically using a forward–backward Runge–Kutta method. In addition, normalized local sensitivity analysis is performed to evaluate the robustness of the optimal strategy to parameter uncertainty.

Results
Numerical simulations compare chemotherapy, immunotherapy, and combined therapy. Chemotherapy is most effective in weak immune environments, while immunotherapy success depends strongly on immune proliferation and lifespan. Combined therapy reduces tumor burden more efficiently than single therapies, although chemotherapy contributes more strongly to tumor elimination. Sensitivity analysis shows that tumor growth rate, immune proliferation, and drug efficacy are the most influential parameters. The optimal strategies remain stable under small parameter perturbations.

Conclusions
The proposed dual-control framework demonstrates how optimal scheduling of chemotherapy and immunotherapy can improve treatment effectiveness while limiting drug usage. The results provide quantitative insight into combination therapy design and highlight the importance of treatment timing and immune dynamics in cancer therapy optimization.